Question

The function $f:Z\to Z$, $f\left(x\right)=\{\begin{array}{cc}0& \text{if}x\text{is odd}\\ \frac{x}{2}& \text{if}x\text{is even}\end{array}$ then $f$ is

• A. surjection but not injection
• B. injection but not surjection
• C. bijection
• D. neither injection nor surjection

Answer 1

The correct option is A surjection but not injection

$f\left(1\right)=f\left(3\right)=0;$ therefore f is not injection
$f\left(2n\right)=n$ for all $n$.

$\therefore$ co-domain of $f=$Range of f $=z$
$\Rightarrow f$ is an surjection
$f$ is a surjection but not injection.